The Need for Open-Ended Math

I’ve been thinking about math a lot lately.
Specifically – I was thinking about how the children in my class were showing me their understanding of math (numbers, measurement, patterns, size/shape, etc) through their play, through their manipulation of the materials they’re working with, through their discussions…

Do I give value to what they’re doing when it’s not ‘math time’?
This is something I am constantly working on; noticing the learning when it’s not assigned ‘learning time’ or when it’s not a time we are all doing the same kind of thing…

Kindergarten, and the whole of early years, is a wonderful time with a bunch of varying abilities, strengths, weaknesses…

I watched a great TED Talk by Jo Boaler (please also watch it) talking about how the myth of having or not having a math brain has stunted our own growth in math, and learning in general (whether we believe we are ‘good at math’ or ‘bad at math’ has an effect on our actual growth and brain development).

I attended a great PD about open-ended math in Beijing at my former school (I can’t remember the name of the man who led the PD, but it got me thinking about the need for open-ended math in every classroom – not only in EY).
The focus was that math needed to include problem solving, focusing on thinking, and explaining thinking…
It needed to be open-ended – multiple right answers.
It needed to be able to be differentiated in itself.
It needed to be interesting, fun, creative…
It needed to be able to be shown in a variety of ways (words, pictures, manipulatives, etc.)

So – we tried it.
And.. we failed.

The reason – we jumped RIGHT into it too soon (pretty sure it was the next day after our PD). The children didn’t have enough experience with these kinds of problems, they didn’t have the thinking skills needed yet, or the independence, and the openness of the problems were too abstract..
Here’s an example:
Miss Alanna wants to buy a dog. She found a dog that was ____ cubes long, but it was too big. She found a dog that was _____ cubes long, but it was too small.
She found a medium sized dog that was _____ cubes long, and it was perfect.

This is a fine problem – but I wouldn’t start with it (…but I did, and it didn’t work: too wordy, too many open slots to fill, not enough background knowledge of size or terms/vocab). I was just too excited I think.. and I also think this would have been fine as a whole class problem, where we could talk about the different ideas – but.. I still wouldn’t use this as our first jump into open-ended problems…

This is an issue I find with a lot of PD – we get inspired, we jump in, it fails or we fail, and we don’t continue.
Instead, we need to lead up to the changes…
I’ve started with open-ended problems with my current class, and they LOVE it.
We’ve started slow, and I chose now because we have a lot of math vocabulary that we’ve built up, we’ve had a solid few months focused on number talks and number sense, we’re currently investigating measurement, and more importantly – they’re ready.
My current class is strong in math – regardless of their needs or language abilities, so this is also a reason why I wanted to focus more on explaining their thinking (in order to give the shy ones a chance to shine in something they’re confident with and strong in, and to give the English language learners a chance to show what they know without feeling like they don’t have the English skills).
My class is used to solving problems on their own, and they’re used to multiple answers to a question (we’ve built this throughout the start of the year, in all areas – not just math).

Here’s how we’ve started.. and I’ll put some examples of where we will head…
Since we are also inquiring into measurement, and continuing with number sense, the problems reflect this (and they also reflect staff that they come in contact with).

Here was our first one,
Mrs Appel loves to swim. She found 3 pools. The smallest pool is 3 cubes long. The medium pool is 10 cubes long. How many cubes long is the biggest pool that Mrs. Appel finds?
We went over this problem as a class, and I used a little toy person to represent Mrs. Appel (our principal) and cubes to show the size of the pools Mrs. Appel found. I tell a story about Mrs. Appel first and about her love of swimming, and how she wanted to find the biggest pool she could find, and that it had to be bigger than the medium sized pool she found.
The children can show their understanding of smaller, bigger, size/scale, number sense/quantity. For this problem – all of the children decided to draw the biggest pool. You can see the child who chose the pool to be 10000000000000 cubes long, has a love of big numbers – and understands size/scale – you can see by how small he drew Mrs. Appel in relation to the giant pool. He also drew racing lanes in the pool 🙂

The next problem we moved on to was a bit more difficult, as it had the children finding a ‘middle’ number between a smaller and larger number.
Mrs. Yan (our TA) wants to buy a medium size dog. She found 3 dogs for sale. The biggest dog is 10 cubes long, the smallest dog is 2 cubes long. Mrs. Yan wants to buy the medium size dog. How many cubes long is it?

Here, the children were more creative in their solving of the problem. I think every child had chose a different number for how long the middle size dog would be, and they all showed their work differently (some drew pictures of the dogs, some drew the cubes to show how long they were). Some recreated the problem using cubes on the carpet, and created their own numbers for smaller, bigger, and medium.
Again, we started on the carpet and I told the story about Mrs. Yan (and we used another little toy doll to represent her), and she joined in our discussion saying the reasons why she couldn’t buy a big or small dog (she said her car was too small to fit a big dog, and she didn’t want a small dog because she was worried about stepping on it).
This showed me the same as the above problem, but the concept of a number in between two numbers. The children need to have a solid understanding of numbers 0-10 to figure this out, as well as the understanding of big, medium, small, more/less.

Since our students were able to answer these without assistance (and in various ways), shows me they have a pretty good understanding of 10, more and less, big/small/medium, and explaining their thinking.

We will move on to another problem that is a bit more challenging, using numbers beyond 10, and then we will move into multiple step problems (more blanks to be filled in!).

My intentions are to have them really thinking about the problem, and how they (individually) could solve it, and that another person might solve it a bit differently, and that’s okay.
These kinds of problems will get more difficult, and more abstract, as we move on. I wanted to start out slowly to really get them thinking this way in math (instead of asking if their answer is right).

**Side note: It’s easy to fall into the trap of worksheets for math where there is a very clear and right answer, it’s easier to assess understanding (not sure if it’s an accurate assessment, but nevertheless), and it’s a more concrete thing. Especially when you google ‘kindergarten math’ and a billion closed-ended worksheets appear and 2 of the top 4 websites are worksheet websites (the other two are closed-ended online games).
Just give this a chance, and see how creative your children can be, and hear their explanations for things.
Also think about how easy it is (and how relaxed you are) to solve a problem when the stress or worry of being right is taken away, and multiple answers are accepted.